# Reconstruction of a sequence generated by a Markov chain - reference request

Let S be a finite sequence of symbols from a finite alphabet, with gaps - that is on some known locations an unknown number of symbols are missing. Assuming that the sequence , including the symbols in the "gaps", is generated by a Markov chain on the alphabet's symbols with an unknown transition matrix, the problem is to design an efficient algorithm that will fill (almost) correctly the gaps , with high probability . More formally , the output sequence must be close (in edit distance , for example) to one of the most probable sequences, given S and the model. The algorithm receive only the sequence S and must output a new sequence S' generated by filling the gaps in S .

This seems to be an important and solvable problem, but I was not able to find any reference . I have some ideas on how to proceed , but I don't want to waste my time on some known result.

I'm imagining the problem can have occurred in bioinformatics or communication theory.